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1 A resonant, self-pumped, circulating thermoacoustic heat exchanger G. W. Swift a) and S. Backhaus Condensed Matter and Thermal Physics Group, Los Alamos National Laboratory, Los Alamos, New Mexico Received May 004; revised 13 August 004; accepted 16 August 004 An asymmetrical constriction in a pipe functions as an imperfect gas diode for acoustic oscillations in the pipe. One or more gas diodes in a loop of pipe create substantial mean flow, approximately proportional to the amplitude of the oscillations. Measurements of wave shape, time-averaged pressure distribution, mass flow, and acoustic power dissipation are presented for a two-diode loop. Analysis of the phenomena is complicated because both the mean flow and the acoustic flow are turbulent and each affects the other significantly. The quasi-steady approximation yields results in rough agreement with the measurements. Acoustically driven heat-transfer loops based on these phenomena may provide useful heat transfer external to thermoacoustic and Stirling engines and refrigerators. 004 Acoustical Society of America. DOI: / PACS numbers: Ud, 43.5.Nm, 43.5.Qp RR Pages: I. INTRODUCTION Stirling s hot-air engine 1, of the early 19th century was one of the first heat engines to use oscillating thermodynamics of a gas in a sealed system. Since then, a variety of related engines and refrigerators has been developed, including Stirling refrigerators, 1,3 Ericsson engines, 4 orifice pulsetube refrigerators, 5 standing-wave thermoacoustic engines and refrigerators, 6 free-piston Stirling engines and refrigerators, 7 and thermoacoustic-stirling hybrid engines and refrigerators also known as traveling-wave systems Combinations thereof, such as the Vuilleumier refrigerator 1 and the thermoacoustically driven orifice pulsetube refrigerator, 13 have provided heat-driven refrigeration. Today, the efficient, mature members of this family of engines and refrigerators are in use in several niche markets, ranging from small cryocoolers with cooling powers below 10 W to large engines e.g., for submarine propulsion with powers near 100 kw. Much of the recent evolution of this family of oscillating-gas thermodynamic technologies has been driven by the search for simplicity, reliability, and low fabrication costs through the elimination of moving parts, especially elimination of moving parts at temperatures other than ambient temperature, without seriously compromising efficiency. For example, the orifice pulse-tube refrigerator lacks the cold piston of previous cryogenic Stirling refrigerators; the free-piston Stirling engine lacks the crankshafts and connecting rods present in previous Stirling engines; and the thermoacoustic-stirling hybrid engine eliminates pistons previously needed. Heat exchangers may offer a second opportunity for dramatic improvement in simplicity, reliability, and low fabrication cost, particularly in engines and refrigerators of high power. All engines and refrigerators must reject waste heat to ambient temperature, and the ambient heat sink is often available as a flowing air stream or water stream. Engines must also accept heat from a heat source at a higher temperature, e.g., a stream of combustion products from a burner. Refrigerators must withdraw heat from a load at lower temperature, which is sometimes in the form of a flowing stream; examples include a stream of indoor air to be cooled and dehumidified and a stream of oxygen to be cooled and cryogenically liquefied. Hence, the typical heat exchanger in these engines and refrigerators must transfer heat between a steadily flowing process fluid stream such as these and the oscillating thermodynamic working gas often pressurized helium. In large, high-power engines and refrigerators, the thermal conductivity of solids is insufficient to carry the required heats without significant temperature differences, so geometrically complicated heat exchangers must usually be used to interweave the process fluid and the working gas, bringing them into intimate thermal contact. A shell-and-tube heat exchanger 14 is typical. In the orientation of Fig. 1a, the working gas oscillates vertically through the insides of the many tubes, while the process fluid flows horizontally around and between the outsides of the tubes. Features specific to oscillating-gas engines and refrigerators impose unfortunate constraints on such heat exchangers as they are scaled up to higher power. Higher power demands more heat-transfer surface area, lest the efficiency suffer. However, tube lengths cannot be increased, because having such tube lengths greater than the oscillatory stroke of the working gas does not effectively transfer more heat. Hence, the obvious approach to scaleup is to increase the number of tubes in proportion to the power, keeping the length and diameter of each tube constant. Such heat exchangers can have hundreds or thousands of tubes, causing expense because many parts must be handled, assembled, and joined and unreliability because many joints must be leak tight. Thermally induced stress poses an additional challenge to reliability when such a geometrically complex heat exchanger is at an extreme tema Electronic mail: swift@lanl.gov; URL: J. Acoust. Soc. Am. 116 (5), November /004/116(5)/93/16/$ Acoustical Society of America 93

2 FIG. 1. a A portion of an oscillating-gas engine or refrigerator, containing a shell-and-tube heat exchanger or other traditional cross-flow heat exchanger. The heat exchanger is below a stack or regenerator, and is above a pulse tube, thermal buffer tube, or open duct. The oscillating flow of the working gas e.g., pressurized helium is vertical throughout. The steady flow of the process fluid e.g., water is horizontal through the heat exchanger. b A portion of an oscillating-gas engine or refrigerator, in which the traditional heat exchanger has been replaced by a single external pipe one wavelength long. Gas diodes at the velocity maxima create mean flow, which steadily transfers heat between the process fluid outside the pipe and the engine or refrigerator to which the pipe ends are attached. c Qualitative plot of volume velocity U as a function of location x in the pipe, at four equally spaced times t during a cycle of the wave. The zero of time phase is when the pressure in the mixing chamber is a maximum. perature, i.e., a red-hot temperature for an engine or a cryogenic temperature for a refrigerator. Another shortcoming of the heat exchangers of oscillating-gas engines and refrigerators is that they often must be located close to one another, simply because each heat exchanger must be adjacent to one end or the other of the nearest stack, regenerator, pulse tube, or thermal buffer tube, and these components themselves are typically short. The practical importance of this shortcoming is easily appreciated by considering the food refrigerator in the typical American kitchen. Its vapor compression cooling technology allows complete flexibility in the geometrical separation of the cold heat exchanger, where heat is absorbed from the inside of the cold box, and the ambient heat exchanger, where waste heat is rejected to the air in the kitchen, typically behind or under the refrigerator cabinet. With vaporcompression technology, not only can these heat exchangers be located freely, but their shapes can be chosen as needed for their circumstances, e.g., to accommodate fan-driven or natural convection, and to fit in and around the desired shape of the cold box or cabinet. In contrast, when one tries to adapt an oscillating-gas refrigerator to this application, the cold heat exchanger and ambient heat exchanger must be very close together, separated only by the regenerator or stack, whose length is typically only several centimeters. Hence, intermediate heat transfer equipment, such as heat pipes or pumped-fluid heat-transfer loops, must typically be used. These add complexity and expense. To explore a new way to circumvent these shortcomings, we have begun to investigate the alternative heat-exchanger scheme illustrated in Fig. 1b. This resonant, self-pumped, circulating thermoacoustic heat exchanger is a single pipe or a few in parallel, which could be bent or coiled compactly, with a length equal to one wavelength of sound in the working gas in the pipe at the frequency of the engine s or refrigerator s oscillation. Both ends of the pipe are attached as branches to the trunk where a traditional heat exchanger would be expected. Oscillations of the gas in the pipe are caused by those in the trunk. Two gas diodes are in the pipe, each located a quarter wavelength from one end of the pipe. These create nonzero mean flow, so the motion of the working gas in the circulating heat exchanger is a superposition of oscillating flow and steady flow. Most of the extensive outside surface area of the pipe is available for thermal contact with the process fluid, which can flow either parallel or perpendicular to the pipe. The mean flow in the loop carries heat between this surface area and the mixing chamber in the trunk. The gas diodes are the key to the heat-transfer circulation in Fig. 1b. These are nonlinear flow elements having different flow resistances for flow in different directions. Gas diodes are typically much less perfect than electronic diodes, often with a ratio of backward and forward flow impedances less than a factor of 10. Gas diodes include the vortex diodes described by Mitchell, 15 the valvular conduit described by Tesla, 16 and the conical and tapered structures called jet pumps in some recent publications. 10,17 The fact that the length of the pipe in Fig. 1b is one wavelength of sound leads to beneficial features, illustrated in Fig. 1c. The gas diodes are located where the oscillating volume velocity 18 is a maximum, so they can create a large time-averaged pressure difference and a large mean flow, as explained more fully below. Meanwhile, the ends of the pipe are locations of minimal oscillating volume velocity, so that connecting the pipe to the trunk perturbs the oscillations in the trunk minimally. Figure 1c illustrates such minimal perturbation with the pipe ends presenting a real impedance to the trunk, but a slightly shorter pipe would add a positive imaginary part to that impedance, which could be useful for canceling unwanted compliance in the trunk. Estimates indicate that the mean flow can be many times larger than the oscillating flow amplitude where the pipe joins the trunk; Fig. 1c illustrates it as.4 times larger. A nonresonant but otherwise similar concept was described by Mitchell 15 for a particular case: the heat exchanger at the ambient end of the pulse tube of an orifice pulse-tube refrigerator. Mitchell replaced this particular heat exchanger and the orifice by a heat-transfer loop containing one or more gas diodes to convert some of the oscillatory 94 J. Acoust. Soc. Am., Vol. 116, No. 5, November 004 G. W. Swift and S. Backhaus: Self-pumped thermoacoustic heat exchanger

3 power in the wave into mean flow of the working gas around the loop. The dissipation in the gas diodes and other acoustic dissipation in the loop serve the dissipative function of the original orifice, and the surface area along the entire path length of the loop serves the function of the original heat exchanger. We undertook the work described here to investigate the main issues that are involved with using the resonant selfpumped loop for heat transfer. The insulated, electrically heated or water-cooled resonant loop described below was attached at both ends to a mixing chamber, where the sound wave was generated by a motor-driven piston and where heat from the electrically heated loop was rejected to watercooled heat exchangers. Thermocouples monitored the temperature rise as the gas circulated through the insulated, heated loop; pressure transducers detected the oscillatory pressure associated with the wave and the time-averaged pressures associated with the mean flow. Most features of the experimental results are explained qualitatively by treating the oscillatory and mean flows as independently superimposed except at the gas diodes. The use of the quasi-steady approximation to analyze interactions of the oscillatory and mean flows throughout the loop, occurring through the Doppler effect and the nonlinearity of turbulent flow resistance, provides a better, reasonably quantitative interpretation of the measurements. An appendix summarizes the most tedious aspects of the application of the quasi-steady approximation to this situation. II. APPARATUS AND PRELIMINARY MEASUREMENTS From among the low-cost gases, we chose argon at a mean pressure p m.4 MPa for this investigation, because its high density gives it a low sound speed and small viscous and thermal penetration depths, which helped to keep the size of the apparatus and the thermal and acoustic powers small enough for easy experimentation. We chose 50 Hz for the operating frequency and cm for the inside diameter of the pipe, for compatibility with a linear motor and piston described below that were available in our lab. The apparatus is illustrated in Fig.. The onewavelength loop shown in Fig. a had a total length of 6.33 m and was made mostly of seamless stainless-steel pipe with an inside diameter D pipe.1 cm. Long-radius elbows and fittings 19 smoothly matched this pipe s inside diameter. A valve at the velocity node halfway around the loop, at the top of Fig. a, could be used to stop and start the mean flow while leaving the acoustics mostly undisturbed. Two geometrically identical sets of piping one with thermocouples, heaters, and thermal insulation, the other cooled by flowing water could be easily interchanged as desired for different types of measurements. For example, equilibration of the temperature profile around the loop required hours with the insulated pipes, but steady state could be reached in a few minutes with the water-jacketed pipes. Gas diodes were located 1/4 and 3/4 of the way around the loop. Each was a machined brass cylinder with fittings brazed onto the ends, as shown in Fig. b. The inside surface was a smooth cone 13.3 cm long, tapering from the.1 cm diameter of the loop piping at its large end to D gd FIG.. Scale drawings of the apparatus. a Overview of the entire apparatus, except for the linear motor and its pressure housing below the bottom. The gas diodes were located 1/4 and 3/4 of the way around the loop of piping, and the valve was halfway around. The locations of most of the thermocouples T and pressure sensors P in the apparatus are shown in this view. The four long, straight runs of pipe each having five thermocouples were wrapped with heating blankets. The entire loop was then wrapped with thermal insulation. However, for some measurements, four alternative straight runs of pipe were used, each having a water jacket instead of the thermocouples, heater blanket, and insulation. The spatial extent of the heating blankets and water jackets is shown in the upper right. b Detailed view of one of the gas diodes, also showing how the fittings come apart. c Detailed view of the drive assembly. The piston, driven by a linear motor, insonified the apparatus. The two optional heat exchangers rejected heat to flowing water at ambient temperature. Temperature and pressure sensors not shown in a are shown here. Bolts holding the drive assembly together are not shown. 1.4 cm diameter at its small end, so the angle between the cone wall and the centerline was The inner edge at the small end was rounded with a radius of.3 mm. These dimensions were chosen to minimize acoustic power dissipa- J. Acoust. Soc. Am., Vol. 116, No. 5, November 004 G. W. Swift and S. Backhaus: Self-pumped thermoacoustic heat exchanger 95

4 tion and to minimize the flow resistance into the small end while simultaneously generating a large time-averaged pressure difference, according to the analysis presented in Sec. III A. Eight piezoresistive pressure transducers 0 were arrayed along the loop as shown in Fig. a, and two more were located in the drive assembly as shown in Fig. c. A lock-in amplifier 1 was used to measure their oscillating voltage amplitudes and phases to obtain the complex pressure amplitude p 1 at each location, and a five-digit voltmeter was used to measure their average voltages for calibration checks and to obtain the second-order, time-averaged pressures p,0 associated in part with the mean flow around the loop. The reference side of the pressure transducer near the piston in Fig. c was always at atmospheric pressure, while the reference sides of the other nine transducers were held at.4 MPa so that the temperature dependence of the transducers gains did not interfere with the measurements of p,0 otherwise, a small drift in temperature would have caused a large drift in average voltage, masking the small voltage changes due to time-averaged pressure. The four transducers closest to the two gas diodes were connected to the loop through thinwalled stainless-steel capillaries 4 cm long and 0.6 mm i.d. so that their temperatures were minimally affected by the wave in the pipe; this feature is discussed in more detail in Sec. III B. Simple modeling of the acoustics of these capillaries and the volumes associated with the transducers indicated that the amplitude drop along the capillary was only 0.1% and the phase shift was only 0.. Figure 3a shows measurements of p 1 for a typical wave with the loop valve closed to prevent mean flow. The corresponding calculated curves in Figs. 3a and 3b were generated straightforwardly by integrating the acoustic momentum and continuity equations with DeltaE, 3 using cone for each gas diode s tapered portion and duct for the rest of the loop. DeltaE s wall roughness factor for turbulence was set at Under the conditions of Fig. 3, turbulence is expected in the gas-diodes cones and near the maxima in volume velocity U 1 in the.1-cm-i.d. pipe, where the gas displacement amplitude over viscous penetration depth is 1 / 830 and the Reynolds-number amplitude N R,1 U 1 D pipe m /S pipe , where D is diameter, m is mean density, S is cross-sectional area, and is viscosity. An impedance at the small end of the cone accounted for the extra dissipation caused by the abrupt area change there, calculated as described near Eq. 19 in Sec. III A. To produce the calculated curves in Fig. 3, the mean pressure, temperature, frequency, and the pressure oscillation amplitude in the mixing chamber were fixed at the experimental values. The agreement between the measured and calculated pressure waves in Fig. 3a is excellent, inspiring confidence that the calculated wave of volume velocity U 1 is accurate, too. The left-right symmetry in Figs. 3a and 3b is nearly perfect: The asymmetry of the two gas diodes orientations has a very small effect on the wave when the valve is closed. Obtaining meaningful measurements of time-averaged pressure required a multi-step procedure. First, the timeaveraged voltages from the pressure transducers were re- FIG. 3. A typical wave, with.4-mpa argon at 48.4 Hz, using the watercooled loop and with the valve closed to enforce zero mean flow, as a function of position around the loop, with x0 andx6.33 m where the loop meets the mixing chamber. Both gas diodes were oriented to encourage flow in the positive-x direction, with their small ends toward smaller x. a Real and imaginary parts of the pressure wave. Points are measured values and lines are calculations. b Calculated real and imaginary parts of the volume-velocity wave. In a and b, the standing-wave components are so much larger than the out-of-phase components that the latter have been multiplied by 10 for clarity. The zero of time phase is chosen so that the pressure oscillations in the mixing chamber are real. c The time-averaged pressure as a function of position, relative to its value in the mixing chamber. Points are measured values and lines are calculations. The two vertical dotted lines are at the small-x, small-diameter ends of the gas diodes. The vertical discontinuity is at the valve. corded while the apparatus was running in steady state. Next, the power to the drive piston was shut off and a valve was opened to equalize the pressures above and below the piston more quickly than would otherwise have occurred through the leakage around the piston. Time-averaged voltages were again recorded from each transducer, and these were subtracted from the earlier measurements. This subtraction accounted for different transducers having different offset voltages. These voltage differences were multiplied by the calibration constants of the transducers to yield p,0 for each transducer. However, these values of p,0 included a large common-mode effect of no interest to us, due mostly to the front-to-back time-averaged pressure difference across the moving piston, generated in part by nonlinear leakage past the piston. Hence, the common-mode part of p,0 was eliminated by subtracting p,0 in the mixing chamber from its value at each of the transducers, yielding p,0. Figure 3c shows a typical set of such results, with p,0 0 in the mixing chamber at x0 and x6.33 m) a consequence of this procedural definition of p,0. 96 J. Acoust. Soc. Am., Vol. 116, No. 5, November 004 G. W. Swift and S. Backhaus: Self-pumped thermoacoustic heat exchanger

5 The calculated curve p,0 (x) shown in Fig. 3c includes both reversible nonlinear effects and irreversible nonlinear effects. The reversible effects arise from what can be loosely described as the acoustic version of Bernoulli s equation, with low time-averaged pressure at locations of high timeaveraged velocity. The well-known lossless expression 4,5 obtained by properly time averaging lossless equations of fluid mechanics p,0 x p 1x 4 m a mu 1 x 4Sx C, 1 where a is the sound speed, with p 1 and U 1 from Figs. 3a and 3b, was used to compute the smoothly varying portions of the curve of Fig. 3c. The two steep portions of the solid curve in Fig. 3c, extending below the bottom of the graph, indicate the contribution of S(x) in the conical tapers of the gas diodes to Eq. 1. The three abrupt steps in the curve of Fig. 3c, two dotted lines at the small ends of the gas diodes and the third no line at the valve, mark changes in the constant C in Eq. 1 from region to region in the apparatus. The small end of each gas diode generates its step in p,0 from a combination of the effect of the step in area in Eq. 1 and the irreversible minor loss. 6 The algorithm used to calculate the magnitude of the minor-loss contribution is described in Sec. III A. The steps in p,0 located at and caused by the gas diodes impose their sum across the valve at x3. m, where the pressure difference of 4 kpa represents the effect that encourages nonzero mean flow as soon as the valve is opened. The experimental values of these irreversible steps shown in Fig. 3c are about 10% smaller than the calculated values. Other than this discrepancy, which is examined in more detail in Sec. III B, the agreement between the experimental and calculated values in Fig. 3c is excellent. The insulated pipes included type K thermocouples and electric-resistance heaters with locations as shown in Fig.. The thermocouples on the loop itself were spot welded to the outside of the pipe and were approximately equally spaced, except for a larger gap across the valve at the top. Wrapped around the outside of the straight sections of the loop were flexible heaters, 7 with power delivered by a variable autotransformer and measured with an electronic power meter. 8 Surrounding the heaters and all the unheated parts of the loop was a layer of flexible foam insulation, cm thick, of the type often used to insulate residential hot water pipes. Measurements with the heaters and thermocouples and with no acoustics showed that the thermal conductance through this insulation was W/ C, about twice the value estimated from the dimensions and the nominal insulating value R5, the excess presumably due to imperfections associated with the valve, fittings, pressure transducers, elbows, etc., and the fact that the heaters covered only the convenient long parts of the loop. To reduce this heat leak, an additional layer of rigid fiberglass insulation of the type often used to cover industrial steam pipes was added to the straight portions, bringing their total diameter to 11.4 cm, and 3 5 cm of soft fiberglass was added around the elbows and valve including the valve handle. This reduced the measured thermal conductance through the insulation to 1.15 W/ C. Figure 4 FIG. 4. Temperature T as a function of position x around the loop for three base line circumstances with the valve closed. Open circles: no acoustic power, 41 W applied with electric heaters, 1.15 W/ C insulation. Inverted triangles: No electric heat, 53 W of acoustic power applied with piston (p 1 15 kpa in the mixing chamber, W/ C insulation. Erect triangles: No electric heat, 14 W of acoustic power applied with piston (p 1 07 kpa in the mixing chamber, W/ C insulation. The points at x0 and x6.33 m are temperatures inside the mixing chamber; other points are temperatures on the wall of the loop. shows measurements of the temperature distribution T(x) for three simple circumstances, giving some indication of the variability in local temperatures resulting from spatial nonuniformity in coverage by the electric heaters, acoustic power dissipation, and thermoacoustic heat pumping. With electric heat and no acoustic oscillations, the thermocouples closest to the unheated portions the gas diodes and the valve are the coolest. With the most intense acoustic oscillations, the thermocouples near the gas diodes, where dissipation of acoustic power is highest, are the warmest. Connecting both ends of the loop to the driving system, at the bottom of Fig. a and shown in more detail in Fig. c, was the mixing chamber. The mixing chamber was an open cylindrical space 10. cm in diameter and 3.18 cm tall, with the loop connections at diametrically opposite locations. In addition to the pressure transducer mounted in its side wall, the mixing chamber contained five thermocouples, extending into the gas itself, axially centered and at various radial and azimuthal positions. A sixth thermocouple extended into the loop 5 cm, on the hot side, i.e., the side delivering mean flow from the loop to the mixing chamber. Water-cooled heat exchangers above and below the mixing chamber, shown in Fig. c, were included when needed to remove heat from the system. Each heat exchanger was a cross-drilled brass block, with 404 holes of.6 mm diameter and 0.6 mm length through which the argon oscillated vertically, and 51 perpendicular holes of the same diameter and various lengths up to 10. cm through which ambienttemperature water flowed horizontally. The oscillations in the system were driven by an oscillating piston of cm diameter, below the lower heat exchanger. The piston was driven in turn by a linear motor, not shown in Fig.. The base plate, piston, and motor came as a complete package from the motor manufacturer, 9 with the motor housing sharing the same gas and mean pressure as the experimental system, so that perfect piston sealing was J. Acoust. Soc. Am., Vol. 116, No. 5, November 004 G. W. Swift and S. Backhaus: Self-pumped thermoacoustic heat exchanger 97

6 unnecessary and the gas exerted no significant time-averaged force on the piston. A mutual-inductance-based linear variable displacement transducer 30 LVDT was mounted on the motor to indicate the piston s mean position m and its complex displacement oscillation amplitude 1. The LVDT was electrically excited at 10 khz and monitored by a lock-in amplifier 1 operating at that frequency. With the lock-in time constant at 3 s, this LVDT lock-in combination was calibrated in situ by driving the motor with enough dc current first of one sign, then of the other so that an opaque edge moving with the piston half blocked the light path of either of two light-emitting diode LED-photodiode sets. The known distance between the two LED-photodiode sets and the lock-in readings at the two positions yielded the calibration of the LVDT lock-in combination. For measurements of m, this lock-in s time constant was kept at 3 s. For measurements of 1, this lock-in s time constant was set at 10 s, and its output was fed to the input of the lock-in used to measure p 1. With the piston moving at 50 Hz at just the right amplitude, comparison of the LVDT signal with the signals from the photodiodes showed that these measurements of 1 are accurate to 1% in amplitude and 1 in phase. To establish more confidence in the measurement of 1, we temporarily replaced the loop, mixing chamber, and heat exchangers with the shortest possible 10-cm-diam cylinder and a dummy load, i.e., a valve functioning as a variable resistance in series with a tank of known volume V. The powers dissipated in the load 31 and delivered by the piston are given by Ė load V Imp p 1,pist p 1,tank, m Ė pist S pist Imp 1,pist 1, 3 where f, f is the drive frequency, is the ratio of isobaric to isochoric specific heats, and the tilde denotes complex conjugation. Figure 5 shows a comparison of these powers for two different pressure amplitudes as the variable resistance was changed. The fact that the observed slopes on this plot are close to unity confirms the accuracy of the measurements of Ė pist and, hence, of 1 and the phase between 1 and p 1. The vertical offsets of the two sets of measurements in Fig. 5 depended on pressure amplitude and were larger than could be accounted for simply by thermal-hysteresis losses on the surface of the short 10-cm-diam cylinder. We assume that the excess is due to oscillatory leakage past the piston. Measurements of this excess power dissipation as a function of amplitude are fit well by FIG. 5. Acoustic power delivered by the piston, Ė pist, as a function of acoustic power delivered to the dummy load, Ė load. Circles, p 1 40 kpa in the mixing chamber. Squares, p kpa in the mixing chamber. The straight line, a guide to the eye, has a slope of unity and an arbitrary vertical offset. Ė leak W/kPa.5 p Some aspects of this leakage dissipation e.g., the dependence of Ė leak on 1 and m ) were not fully explored at the detailed level of a few watts. Greater care was not justified, because other dissipative effects in the mixing chamber, such as the effect of amplitude-dependent turbulence on the thermal-hysteresis losses on the surfaces of the heat exchangers, are not understood at that level of detail. Overall, we estimate that the measurements of the acoustic power delivered to the mixing chamber by the piston have an uncertainty of about % due to sensor-calibration uncertainties plus a few watts due to this leakage dissipation. During the course of the measurements, with temperature often a significant function of location x around the loop and with mean flow often nonzero, an easy experimental definition of loop resonance was needed so that the drive frequency could be chosen without having to map out the entire wave as shown in Fig. 3a. We decided to use the condition Re 1 /p 1,pist 0 m/kpa to define resonance, where p 1,pist is the complex pressure oscillation amplitude above the piston and the numerical value of 0 m/kpa is obtained from the calculated compliance of the mixing chamber, heat exchangers, and adjacent 10-cm-diam spaces in the driver assembly. The numerical value varied slightly with amplitude, because m varied slightly. For left-right symmetry in the loop, Eq. 5 implies that the impedances of both branches from the mixing chamber to the loop are real, an unambiguous definition of loop resonance frequency. For more general, asymmetrical conditions, Eq. 5 implies that the sum of the imaginary parts of the inverse impedances of the two branches is zero, so that if one end of the loop looks inertial the other must look compliant. Before recording any other data, measurements of 1 and p 1,pist were obtained, Re 1 /p 1,pist was computed, f was adjusted, and the process was repeated until Eq. 5 was satisfied. This typically took only 1 min to reach a resonance frequency precise to 0.01 Hz. However, the accuracy of the resonance frequency determined in this way was in doubt 0.5 Hz. The numerical value used in Eq. 5 was uncertain because of uncertainty in how much of the calculated compliance volume, which included the heat-exchanger passages, should be regarded as isothermal instead of adiabatic. The more accurately known value of 17.4 m/kpa was used 5 98 J. Acoust. Soc. Am., Vol. 116, No. 5, November 004 G. W. Swift and S. Backhaus: Self-pumped thermoacoustic heat exchanger

7 for measurements when the heat exchangers were omitted. With the water-cooled pipes in use, the experimental resonance frequency defined in this way decreased when the valve was opened at fixed acoustic amplitude. The decrease varied from 0.5% at low amplitude to 0.7% at high amplitude. Most of this change can be attributed to the 0.4% increase in length of the loop as the passage inside the ball of the valve became part of the acoustic path. The experimental resonance frequency increased quadratically with acoustic amplitude, by 0.6% from low amplitude to p 1 40 kpa when the valve was closed and by 0.8% when the valve was open. How much of this rise is attributable to temperature rise in the gas is unknown. Note that about a third of the acoustic power dissipation in the loop occurs in the gas diodes, which are at velocity maxima of the wave where a change in temperature has the largest effect on the resonance frequency. In the calculations plotted in Fig. 3 and throughout the rest of the paper, the experimental frequency was used in the calculations. Forcing the calculations to mimic the experimentally defined resonance i.e., to zero the sum of the calculated imaginary parts of the impedances of the two ends of the loop at the mixing chamber could be accomplished either by setting the calculation s frequency 1% above the experimental resonance frequency or by reducing the calculation s temperature by % to reduce the sound speed by 1%. However, this alternative calculation scheme made little difference in other calculated results. III. CIRCULATING MEAN FLOW A. Theory In Fig. 3c, the step in p,0 across the valve shows that the loop is ready to deliver nonzero time-averaged mass flow Ṁ in the positive-x direction as soon as the valve is opened. In this subsection, analysis directed toward understanding such time-averaged effects is presented. This analysis is based on simple assumptions, but nonetheless captures many of the experimental features described below. The quasi-steady approximation is central: We proceed as if the results of steady-flow analysis can be applied at each instant of time without memory of recent past history. The irreversible part of the turbulent-flow pressure difference p across any lumped element, including minor-loss components such as gas diodes, is conventionally 3 expressed using the minor-loss coefficient K pku /, where is the gas s density and u is its velocity at a reference location in the lumped element. Hence, for superimposed steady and oscillatory flows, the irreversible part of the time-averaged pressure difference developed across each component due to the time-dependent flow through it can be estimated using 6 p / ptdt 0 /t S 0K 1 t0 U 1sin tṁ/ dt /t 0K 1 /t0 U 1sin tṁ/ dt, 8 where K and K are the minor-loss coefficients for the two directions of flow through the component, S is the area on which the K s are based conventionally, the smallest crosssectional area of the component, t is time, and t 0 is the time at which the volume velocity crosses zero, i.e., t 0 satisfies U 1 sin t 0 Ṁ/0. If Ṁ/U 1, no solution for t 0 exists, and either the K or K integral is carried out from 0 to /. Otherwise, the zero crossing with /t 0 0is chosen. Equation 8 is a straightforward extension of the discussion near Eq in Ref. 33. Our choice of notation for the steady flow calls for some explanation, because no choice seems completely satisfactory. Results below show that Ṁ is approximately proportional to the first power of the amplitude of the wave and is comparable in size to m U 1, but a subscript 1 would be misleading because it would improperly suggest that Ṁ is complex or oscillatory. A subscript m would inappropriately suggest that this steady flow exists in the absence of the sound wave just as p m and m exist in the absence of the wave. We have chosen to use Ṁ, without subscripts, because it is simple and because the most reliable measurements described below detect mass flow, not volume flow. The mass flow Ṁ in this paper should not be confused with the much smaller, second-order time-averaged mass flow Ṁ 1 Re 1 Ũ 1 m U,0 in previous work. 10,11,33 Assuming that all variables except t itself are independent of time, performing the integrals in Eq. 8 yields the irreversible second-order time-averaged pressure difference p,0 K Ṁ / m S m U 1 /4S for 1 9 mu 1 8S K K 1 K K K K 1 sin 1 31 for 1 10 K Ṁ / m S m U 1 /4S for 1, 7 11 where the flow-rate ratio Ṁ/ m U 1. Our gas diodes operate with 1, so that the flow passes through zero twice during each cycle. When 1, the flow is unidirectional. Signs have been chosen so that positive p discourages positive Ṁ and K corresponds to flow in the x direction. To obtain p gd,0 for our gas diodes geometry from Eq. 10, we use the well-established Borda Carnot expression 34 K 1S gd /S pipe 1 for minor-loss flow through the abrupt expansion from S gd to S pipe to obtain K We combine equations and a J. Acoust. Soc. Am., Vol. 116, No. 5, November 004 G. W. Swift and S. Backhaus: Self-pumped thermoacoustic heat exchanger 99

8 chart from Ref. 3, for an abrupt contraction with a rounded edge plus a term accounting for radial nonuniformity in a short conical expander, to obtain K These quasisteady estimates and the entire quasi-steady approach introduced in Eq. 8 should be accurate for oscillatory flow if the Reynolds number is sufficiently high and if the gas displacement is sufficiently large. For the conditions of Fig. 3, N R,1,gd U 1 D gd /S gd , but 1 /D gd is only 9, the latter condition perhaps not extreme enough for complete confidence in the quasi-steady approximation. 35,36 Equations 9 11 with K K still depend on, so apparently the oscillatory flow affects the time-averaged pressure drop across symmetrical turbulent components as well as across the asymmetrical gas diodes. Setting K K K in Eqs yields p,0 K m Ṁ/ m S pipe 1 1 for 1 13 K m Ṁ/ m for 1. S pipe 1 sin This multiplicative enhancement of the steady-flow pressure drop across a symmetrical turbulent component grows from near unity at small U 1 though 3/ at U 1 Ṁ/ m toward a linear asymptote of 4 m U 1 /Ṁ at large U 1. The origin of this increase is the fundamentally nonlinear nature of turbulent flow resistance: If the flow is described by p(t) RU(t) n with n1, then the extra pressure difference caused by an increment of mass flow above the average value Ṁ is not canceled when an equal decrement below the average occurs half a cycle later. In the loop under investigation here, opposing the pressure differences of Eq. 10 that are generated at the two gas diodes is the time-averaged pressure gradient throughout the rest of the loop due to Ṁ flowing around the loop. To estimate this pressure gradient, one might expect 37 that standard equations of fluid mechanics, such as dp dx f m M D pipe Ṁ/ m S pipe, 15 could be used for the mean flow through the uniform-area parts of the loop, where D pipe is the pipe diameter and f M is the Moody friction factor, which is given in most fluid mechanics textbooks e.g., Ref. 6. Some of the calculated curves presented below rely on this simple treatment of the mean flow in the loop. However, Eq. 15 describes nonlinear, turbulent flow, so the arguments presented in the previous paragraph suggest that Eq. 15 could significantly underestimate the mean-flow resistance when oscillatory flow is superimposed. Furthermore, the Moody friction factor f M depends on velocity, so the constant-k analysis leading to Eqs. 13 and 14 is not suitable. The appendix presents a derivation of expressions similar to Eqs. 13 and 14 taking the velocity dependence of f M into account; the result is Eqs. A9 and A10. To Eq. 15 or Eqs. A9 and A10 for the timeaveraged pressure gradient along the uniform-diameter portions of the pipe are added two pressure drops due to drag on the mean flow by the conical surfaces of the two gas diodes, 3,38 four pressure drops due to steady-flow minor loss corrections in the four 90 elbows, 34 and one pressure drop due to steady-flow minor loss at the exit from the pipe to the mixing chamber; all of these may also be increased by nonzero U 1 according to the multiplicative enhancement of Eqs. 13 and 14. These seven pressure drops typically summed to about 50% of that of the uniform-diameter portions. Setting the sum of the forcing pressures of the two gas diodes equal to the sum of all these opposing steady-flow pressure drops allows one to find Ṁ. This must be done numerically because of the complicated nature of the equations. Qualitatively, Eq. 10 shows that p gd,0 is proportional to the square of the wave amplitude and Eq. 15 shows that the opposing pressure gradient in the loop is nearly proportional to the square of the mean flow, so one can expect that the mean flow is roughly proportional to the wave amplitude. Again using the quasi-steady approximation, the acoustic power consumed by a turbulent lumped element such as a gas diode due to the oscillatory component of the flow through it can be estimated using 39 Ė / ptu1 sin t dt 0 S t0 /t 0K 1 U 1 sin t dt /t0 Ṁ/ U 1 sin t dt. U 1sin tṁ/ /t 0K 1 U 1sin t This is a straightforward extension of the discussion near Eq in Ref. 33. Again assuming that variables other than t itself are independent of time, performing the integrals yields 39 ĖK m U 1 3 mu 1 3 3S K K S for sin1 K K K K 3 4 for 1 19 m U 1 3 K S for 1. 0 We then use p 1 Ė/U 1 as an estimate of the first-order pressure difference caused by irreversible processes in such components. As in Eqs. 9 and 10, note the nontrivial dependence in Eqs. 18 0, even if K K, indicating that acoustic- 930 J. Acoust. Soc. Am., Vol. 116, No. 5, November 004 G. W. Swift and S. Backhaus: Self-pumped thermoacoustic heat exchanger

9 power dissipation in a symmetrical turbulent component is increased by superimposed steady flow. This increase is given by a multiplicative factor sin1 for 1, 1 3 for 1 4 if the component s K is independent of velocity. The Moody friction factor f M depends on velocity, so a more complicated analysis must be used to estimate the dissipation of acoustic power in the straight portions of the pipe in the presence of superimposed steady flow. The appendix presents a derivation of expressions similar to Eqs taking the velocity dependence of f M into account; the result is Eqs. A11 and A1. B. Experiments with nonzero mean flow To quantitatively test for Ṁ0 as predicted in the previous subsection, we operated the loop with the valve open. The effect of the mean flow on wave shape and p,0 provided indirect measurements of Ṁ, and the heat carried by Ṁ provided a nearly direct measure. The water-cooled pipes were most convenient for measurements of wave shape and p,0, because rapid equilibration to steady state enabled rapid data taking and because the spatial uniformity of temperature yielded more confidence in calculations of the wave. Measurements with the heated and insulated pipes were used to detect heat carried by Ṁ. With the valve open, the loop has the topology of a constricted annular resonator driven at one point, whose complicated behavior has been described by Muehleisen and Atchley. 40 Without constrictions or dissipation, the modes of such a resonator are degenerate: With the length of the loop equal to one wavelength, standing waves of any spatial phase and traveling waves of either direction satisfy the wave equation and share a single resonance frequency. Reference 40 shows that an area constriction eliminates the degeneracy, splitting the resonance into a high standing-wave mode and a low standing-wave mode. In the low mode, a velocity antinode is centered on the constriction, so the resonance frequency is reduced because of the increased inertance of the constriction. In the high mode, a pressure antinode is centered on the constriction, so the resonance frequency is increased by the reduced compliance of the constriction. In the present experiment, the gas diodes are constrictions, and the low mode is desired. The resonance condition described near Eq. 5 selects the low mode. Only Hz higher in frequency, a weak local maximum in p 1 at the gas diodes suggests the presence of the high mode. The steps at the gas diodes in the calculated curves for Imp 1 and p,0 in Figs. 3a and 3c are based on Eqs. 19 and 10, respectively, with SS gd and Ṁ0. These steps, and all other features of the curves, are in qualitative agreement with the measured pressures. Figure 6 shows corresponding measurements and calculations for Ṁ0, with the valve open at the top of the loop. The overall character of the wave is still that of a standing wave, as indicated by the cosine shape of Rep 1 in Fig. 6a, for which measured and calculated values are in good agreement. However, Imp 1 changed dramatically when the valve was opened. The solid lines in Figs. 6a and 6b represent the results of calculations using DeltaE, as described near the beginning of Sec. II and with the steady-flow phenomena calculated assuming that the mean and oscillatory components of the velocity are independent except at the gas diodes where they are linked through Eqs. 10 and 19. The dramatic disagreement between the experimental Imp 1 and this calculation motivated us instead to model the Doppler effects for which DeltaE cannot yet account of opposite signs in the nearly equal counterpropagating traveling-wave components of the standing wave. In the Doppler model, implemented in a spreadsheet, we used p 1 Ae ik x Be ik x, U 1 is m k 1im D ik Ae ik x ik Be ik x, 3 4 aṁ/s 1im 1 D, 5 where k are the wave vectors for waves traveling with and against 41 the mean flow, A and B are the pressure amplitudes of those waves, is the thermal penetration depth, and i 1. With (1) /D pipe 0.005, the boundarylayer approximation used in Eqs. 3 5 is well justified. In the spreadsheet, the turbulence correction factor m is calculated according to the algorithm used in DeltaE 3 and described near Eq. 7.6 in Ref. 33, but with an additional correction due to the superimposed steady flow given in Eqs. A15 and A16. For laminar flow, m1; for turbulent flow, m1. In the Doppler-model spreadsheet, each of the four long runs of pipe was subdivided into eight pieces, and each gas-diode cone into two pieces; further subdivision did not change the results. Equations 3 and 4 were used for wave propagation in each such piece, with continuity of p 1 and U 1 joining solutions between pieces. As in the DeltaE calculation, Eq. 19 was used for the minor-loss contribution to p 1 across the gas diodes and Eqs. 10, A10, and the discussion in the paragraph following Eq. 15 were used to obtain Ṁ. Complex U 1 at x0 was used as an adjustable parameter to ensure that complex p 1 (0)p 1 (6.33 m). The calculation s gas temperature, which was constant along the pipe, could also be adjusted slightly above the experimental value to enforce ImU 1 (0)ImU 1 (6.33 m), corresponding to our operational definition of resonance, but this adjustment had a negligible effect. The result, shown as the dashed lines in Figs. 6a and 6b, displays Rep 1 and Imp 1 in agreement with the experimental values. The agreement between this calculation and the measurements strongly suggests that the Doppler effect is primarily responsible for the large values of Imp 1. To consider this effect in more detail, notice that the measured Imp 1 in Fig. 6a is not left-right antisymmetric; the depth of the minimum near x4.8 m exceeds the height J. Acoust. Soc. Am., Vol. 116, No. 5, November 004 G. W. Swift and S. Backhaus: Self-pumped thermoacoustic heat exchanger 931

10 FIG. 7. The part of Imp 1 near the gas diodes that is associated with Ṁ 0, as a function of the amplitude of the wave. The horizontal axis is the amplitude of the pressure oscillation in the mixing chamber, raised to the 3/ power. The four sets of symbols measurements and the four curves calculations correspond to the four pressure transducers closest to the two gas diodes. The measurements were taken with the water-cooled pipes. FIG. 6. a c A typical wave, with.4-mpa argon at 48.1 Hz, using the water-cooled pipes and with the valve open to allow Ṁ0. The two gas diodes were oriented to encourage flow in the positive-x direction, with their small ends toward smaller x. a Real and imaginary parts of the pressure wave as a function of position around the loop. Points are measured values and lines are calculations. b Calculated real, imaginary, and mean parts of the volume velocity. In a and b, the standing-wave components are so much larger than the out-of-phase components that the latter have been multiplied by 3 for clarity. The zero of phase is chosen so that the pressure oscillations in the mixing chamber are real. c The time-averaged pressure as a function of position, relative to its value in the mixing chamber. Points are measured values and lines are calculations. The two vertical dotted lines are at the small-x, small-diameter ends of the gas diodes. d Temperature as a function of position around the loop, with the insulated pipes and the W/ C insulation, under conditions similar to that of a c. Filled symbols are temperatures measured on the metal surface of the loop and open symbols are temperatures measured in the gas. Gas temperatures at x0 and x6.33 m are those at five locations in the mixing chamber. Circles, no electric heat. Squares, 79 W electric heat. Erect triangles, 167 W electric heat. Inverted triangles, 5 W electric heat. The curves represent calculated estimates corresponding to the erect triangles. of the maximum near x1.6 m. This downward shift resembles that of the Ṁ0 data in Fig. 3a, where Imp 1 0 is associated with the flow of acoustic power into the loop from both ends. To more clearly distinguish the effect of Ṁ0 from that of the acoustic power flow, in Fig. 7 we display the difference between Ṁ0 and Ṁ0 measurements of Imp 1 at the four pressure transducers closest to the two gas diodes, at five different pressure amplitudes. The subtraction brings the results at the four transducers into agreement with one another, and plotting these against the 3/ power of the wave amplitude yields a reasonably straight line. We do not understand why Ṁ0 should contribute to Imp 1 in proportion to the 3/ power of the wave amplitude. Our spreadsheet-based Doppler calculations, also displayed in Fig. 7, show the difference between Imp 1 when Ṁ0 and Ṁ0 more nearly proportional to the square of the wave amplitude. The difference between calculated and measured Imp 1 in Fig. 7 might simply be due to Ṁ depending on physics that we do not understand. However, one other candidate is apparent: Both the calculations and the measurements for Ṁ0 displayed a remarkably strong dependence of Imp 1 on frequency as frequency was changed away from resonance. Both rose with frequency; measured values rose about 30% per Hz and calculated values about 40% per Hz. Hence, some of the difference between calculations and measurements in Fig. 7 could be a small amplitude-dependent error or ambiguity about what really constitutes resonance frequency or what gas temperature should be used in the calculations. With such an imperfect understanding of Imp 1, we cannot be confident in the accuracy of ReU 1 in Fig. 6b. However, Rep 1 appears to be well understood, and hence we should have confidence in the calculation of ImU 1 d Rep 1 /dx, which is the dominant part of the volumevelocity wave. The time-averaged pressure difference across the gas diodes and the resulting mean flow are due to U 1 93 J. Acoust. Soc. Am., Vol. 116, No. 5, November 004 G. W. Swift and S. Backhaus: Self-pumped thermoacoustic heat exchanger